sqrt(2)*sin(x)=1 (уравнение)

Найду корень уравнения: sqrt(2)*sin(x)=1

Вы ввели

  ___           
\/ 2 *sin(x) = 1

$$\sqrt{2} \sin{\left(x \right)} = 1$$

Подробное решение

Дано уравнение
$$\sqrt{2} \sin{\left(x \right)} = 1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на sqrt(2)

Ур-ние превратится в
$$\sin{\left(x \right)} = \frac{\sqrt{2}}{2}$$
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Или
$$x = 2 \pi n + \frac{\pi}{4}$$
$$x = 2 \pi n + \frac{3 \pi}{4}$$
, где n - любое целое число

Быстрый ответ

     pi
x1 = --
     4 

$$x_{1} = \frac{\pi}{4}$$

     3*pi
x2 = ----
      4  

$$x_{2} = \frac{3 \pi}{4}$$

Сумма и произведение корней

сумма

    pi   3*pi
0 + -- + ----
    4     4  

$$\left(0 + \frac{\pi}{4}\right) + \frac{3 \pi}{4}$$

=

pi

$$\pi$$

произведение

  pi 3*pi
1*--*----
  4   4  

$$\frac{3 \pi}{4} \cdot 1 \frac{\pi}{4}$$

=

    2
3*pi 
-----
  16 

$$\frac{3 \pi^{2}}{16}$$

Численный ответ

x1 = 69.9004365423729

x2 = 128.019900633784

x3 = -30.6305283725005

x4 = 7.06858347057703

x5 = 38.484510006475

x6 = -237.975643509427

x7 = 2.35619449019234

x8 = -16.4933614313464

x9 = -4340.8956490977

x10 = -80.8960108299372

x11 = -93.4623814442964

x12 = -85.6083998103219

x13 = -10.2101761241668

x14 = -2179.47990342792

x15 = -54.1924732744239

x16 = -11.7809724509617

x17 = 57.3340659280137

x18 = 19.6349540849362

x19 = -68.329640215578

x20 = 101.316363078271

x21 = -29.0597320457056

x22 = -79.3252145031423

x23 = 15074.1469500872

x24 = -3.92699081698724

x25 = 71.4712328691678

x26 = 90.3207887907066

x27 = 21.2057504117311

x28 = -24.3473430653209

x29 = 52.621676947629

x30 = 44.7676953136546

x31 = 58.9048622548086

x32 = 77.7544181763474

x33 = 46.3384916404494

x34 = 27.4889357189107

x35 = -18.0641577581413

x36 = 82.4668071567321

x37 = 32.2013246992954

x38 = 40.0553063332699

x39 = -118.595122673015

x40 = -73.0420291959627

x41 = 51.0508806208341

x42 = -74.6128255227576

x43 = -66.7588438887831

x44 = 76.1836218495525

x45 = 25.9181393921158

x46 = -186.139364725195

x47 = -1160.03308733803

x48 = -5.49778714378214

x49 = 13.3517687777566

x50 = -60.4756585816035

x51 = -98.174770424681

x52 = 63.6172512351933

x53 = 2151.20556954561

x54 = 33.7721210260903

x55 = -91.8915851175014

x56 = -87.1791961371168

x57 = -36.9137136796801

x58 = 8.63937979737193

x59 = 14.9225651045515

x60 = 65.1880475619882

x61 = -41.6261026600648

x62 = -99.7455667514759

x63 = -55.7632696012188

x64 = -35.3429173528852

x65 = -49.4800842940392

x66 = 96.6039740978861

x67 = -47.9092879672443

x68 = 95.0331777710912

x69 = -22.776546738526

x70 = 0.785398163397448

x71 = -62.0464549083984

x72 = 88.7499924639117

x73 = 84.037603483527

x74 = -43.1968989868597

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